Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Interpretation of $S$-ideal class group

I have a problem understanding the interpretation of the ideal class group in the case of restricted ramifiction. Let $k$ be a number field and $S$ a set of primes of $k$. Then $k_S$ denotes the ...
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0answers
105 views

Question about why $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$.

I'm trying to follow a solution I'm reading. The idea is to prove $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$ when $n=1$ or $n=p=2$. The solution is as follows: If $x^{p^n}-x+1$ is ...
1
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1answer
249 views

Galois group of an irreducible cubic polynomial without using the discriminant

Let $f\in\mathbb Q[x]$ be irreducible of degree $3$. Since the Galois group $G$ of $f$ is a transitive subgroup of $S_3$, it is either $S_3$ or $A_3$. Those two possibilities are easily ...
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1answer
50 views

Make Galois extension large enough to let Galois group act trivially on module

Let $K|k$ be a finite Galois extension of number fields, inside a given "maximal" (infinite) Galois extension $k_S$ of $k$. Let $G = Gal(k_S | K)$ denote the Galois group and let $A$ be a ...
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0answers
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Generalized fundamental theorem of Galois theory

In the generalized fundamental theorem of Galois theory, theorem says there is a one-to-one correspondence between the set of all intermediate fields of extension and the set of all $\textbf{closed}$ ...
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0answers
69 views

A Problem about Galois Extension

This is a basic question about Galois Extension, but I want some details about it. Let $F$ be a splitting field over $\mathbb Q$ the polynomial $x^8-5\in\mathbb Q[x]$. Recall that $F$ is the subfield ...
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3answers
101 views

Automorphisms on a field $F$

I am trying to understand this proposition with respect to algebraic closures of a field $F$ Prop: If $F$ is a finite field, then every isomorphism mapping $F$ onto a subfield of an algebraic closure ...
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0answers
204 views

Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
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1answer
74 views

composition of a $\mathbb{Z}_p$-extensions and Hilbert class field extension (Iwasawa)

I am reading Iwasawa's article On $\Gamma$-extensions of algebraic number fields. In paragraph 7.3 : " We now take as M the maximal unramified abelian $p$-extension of $L$ in $\Omega$ , i.e. the ...
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0answers
50 views

question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
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1answer
49 views

Uniqueness of $p^{th}$ powers in characteristic $p$

The other day, my undergrad Galois theory professor used the fact that in char $p$, $p$th powers exist and are unique. How can one understand why uniqueness holds? Thanks
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$\rho=e^{\frac{2\pi i}{21}}$, Prove $\rho+\rho^4+\rho^{16}$ is constructible

Let $\rho=e^{\frac{2\pi i}{21}}$. Prove that the number $a=\rho+\rho^4+\rho^{16}$ is constructible using a compass and a straightedge. A partial solution was to define a $\mathbb{Q}$-automorphism ...
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3answers
435 views

Applications of Galois theory for topology

Is there any applications of Galois theory in topology? I already have learned Galois theory, and applied it in algebra. Can I get solution of some big problem about topology using Galois theory? ...
8
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2answers
162 views

The angle $168^\circ$ is constructible

Prove that the angle $\theta=168^\circ$ is constructible using a straightedge and a compass. It is enough to show that the number $\cos\theta$ is constructible, and WolframAlpha gave ...
4
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2answers
223 views

If Gal(K,Q) is abelian then |Gal(K,Q)|=n

Let $f(x)\in \mathbb Q[x]$ irreducible of degree $n$ and $K$ its splitting field over $\mathbb Q$. Prove that if $\operatorname{Gal}(K/\mathbb Q)$ is abelian, then $|\operatorname{Gal}(K/\mathbb ...
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1answer
65 views

No field extension is “degree 4 away from an algebraic closure”

I have seen this problem asked by another user but it isn't completely solved in the answers. I'm trying to do it, but I can't. Question: Suppose $[L:K]=4$ and $charK≠2$ and $L$ is algebraically ...
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1answer
92 views

A problem in Galois Theory

While reading algebraic number theory, I came across the following statement: Let $K$ be a galois extension over $\mathbb{Q}$ and $H$ be the Hilbert class field (maximal unramified abelian extension) ...
3
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1answer
110 views

Splitting field of x^3-2 as a simple extension

Is there any elegant way to show that $\mathbb Q(\sqrt[3]{2}, w)=\mathbb Q(\sqrt[3]{2}+w)$, where $w=e^{i\frac{2\pi}{3}}$. I was thinking to show that $ 9+9 x+3 x^3+6 x^4+3 x^5+x^6$ is the minimal ...
12
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1answer
238 views

A result of van der Waerden says Galois theory “needs” incomputable sets - what does this mean, exactly?

I happened across the recent arXiv paper Transfinite Recursion In Higher Reverse Mathematics, and the introduction begins: The question "What role do incomputable sets play in mathematics?" has ...
4
votes
3answers
147 views

give an example of algebraic numbers $\alpha, \beta$ such that…

Question is to find algebraic numbers $\alpha, \beta$ such that : $$[\mathbb{Q}(\alpha):\mathbb{Q}]>[\mathbb{Q}(\beta):\mathbb{Q}]>[\mathbb{Q}(\alpha\beta):\mathbb{Q}]$$ It is not so difficult ...
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4answers
759 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
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2answers
1k views

Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
8
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1answer
166 views

Question about inverse Galois problem

I have a question... if for every finite simple group, we can construct a Galois extension over $\mathbb{Q}$ with that Galois group, does it follow that we can construct Galois extensions over ...
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0answers
69 views

Proof that $p(z)^2=a^2$ always has a nonreal solution.

Let $p(z)$ be a nonconstant integer polynomial of degree $n$ such that $p(0)=0$ and let $a$ be a nonzero real number. It seems that $$p(z)^2=a^2$$ Always has a nonreal solution (in $z$) if ...
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2answers
205 views

Splitting Field of $x^6-6x^3+7$

Given the polynomial $p(x)=x^6-6x^3+7 \in \mathbb Q$, find its splitting field $\mathbb F\subset \mathbb C$ and the Galois group of the extension $\mathbb F /\mathbb Q$. In fact, the exercise asked to ...
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3answers
27 views

About concrete finit Galois extension.

I'm trying to solve this problem, but I don't know what I can do. Let $f(x)=x^3+2x+1$ over $\mathbb{F_5}$. Let $E$ the decomposition field of $f$ over $\mathbb{F_5}$ and $\alpha$ a root of $f$ in ...
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1answer
53 views

Gauss lemma in UFDs

Let $A$ be a UFD, and $f\in A$ a square-free element. Define the integral domain $B:=A[z]/(z^2-f)$, and consider a monic polynomial $F(T) \in B[T]$ such that $F(\alpha) = 0$ for some $\alpha \in ...
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1answer
307 views

Galois group of a quartic

Let $x^4+ax^2+b$ in $K[x]$ (with char $K\neq $2) be irreducible with Galois group $G$. (a) If $b$ is a square in $K$, then $G = \mathbb{Z}_2\times\mathbb{Z}_2$. (b) If $b$ is not a square in $K$ ...
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2answers
103 views

Why $S_4$ has no transitive subgroups of order 6?

I know that every transitive subgroup of $S_4$ have to be order divisible by 4, but i should solve this with Galois Theory. I think this theorem can be usefull: Theorem 4.2. Let K be afield and f in ...
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1answer
48 views

Help with Polynomial Roots Problem

Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$. Suppose I want to find when there is $c\in\mathbb{R}$ such that $$p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } ...
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1answer
390 views

Intermediate fields of a field extension

Let $L := \Bbb Q(\sqrt 5,\sqrt 7).$ We have proved that $L/\Bbb Q$ is galois. I have to find all the intermediate fields of $L/\Bbb Q$. So far, I have found $\Bbb Q(\sqrt 5)$, $\Bbb Q(\sqrt 7)$, ...
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1answer
123 views

Question on the proof of existence of splitting fields for a family of polynomials

I have a question regarding the following well known result: Let $C\subseteq K[x]$ be a family of polynomials. We know that $C$ possesses a splitting field over $K$. The proof I am reading goes like ...
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1answer
274 views

Beating Gauss on Irreducibility of Cyclotomic Polynomials?

I am considering how to provide an alternative proof of the lemma used in the proof that $\Phi_n$ is irreducible: Lemma: If $\Phi_n=f_1 f_2\cdots f_r$ is a factorization into monic irreducible ...
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1answer
335 views

Separability and tensor product of fields

Is it true that a finite degree field extension $L/k$ is separable if and only if $L\otimes_{k}L$ is a reduced $L$-algebra? Surely the "only if" part is true because if the extension is ...
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4answers
93 views

Let $S=\big\{\sqrt[n]{3}\colon n\in \mathbb{N}\big\}$. Is the extension $\mathbb{Q}[S]\colon\mathbb{Q}$ algebraic?

A field extension $L\colon K$ is algebraic if every element in $\alpha \in L$ is algebraic over $K$. An elemenet $\alpha \in L$ is algebraic over $K$ if there exists a polynomial $f \in K[x]$ such ...
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0answers
99 views

Systematically describing the Galois Group and Intermediate Fields

In an exercise in the textbook you are asked to describe the Galois Group and the intermediate fields of the extension $$ L=\newcommand{\Q}{\mathbb Q}\Q(\sqrt 2,\sqrt 3)\supset\Q $$ I have noted that ...
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5answers
845 views

Ramanujan-type trigonometric identities with cube roots, generalizing $\sqrt[3]{\cos(2\pi/7)}+\sqrt[3]{\cos(4\pi/7)}+\sqrt[3]{\cos(8\pi/7)}$

Ramanujan found the following trigonometric identity \begin{equation} \sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}= ...
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1answer
50 views

Find a polynomial whose splitting field is $\mathbb{Q}[\alpha,i]$

Let $f(x)=x^{3}-3x+1$ and let $\alpha$ be a root in $f$. i) Show that the polynomial $f$ is irreducible in $\mathbb{Q}[x]$. ii) Show $\alpha^{2}-2$ is a root of $f$ as well, and show that all roots ...
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1answer
53 views

Permutation of roots for Galois group with six elements

We know that $\mathbb{Q}(\sqrt[3]{2},i\sqrt{3})$ is the splitting field of $x^3-2$ over $\mathbb{Q}$, and $[\mathbb{Q}(\sqrt[3]{2},i\sqrt{3}):\mathbb{Q}]=6$. Now, consider an element $\alpha$ in the ...
2
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1answer
58 views

Calculating polynomials in a Galois field

I'm in $\text{GF}(8) = \text{GF}(2^3)$ and have an irreducbile polynomial $p(x) = x^3 + x + 1$, then $\text{GF}(8) = \mathbb{Z}_2[x]/\langle p(x) \rangle$ . Now I want to multiply $2$ elements of ...
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2answers
483 views

Galois group command for Magma online calculator?

I need to test if a family of 7th deg and 13 deg equations are solvable. I'm new to Magma, so my apologies, but what would I type in, http://magma.maths.usyd.edu.au/calc/ to determine the Galois ...
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1answer
116 views

Being Galois stable under completion?

Let $R$ a Dedekind Domain, $K = \mathrm{Frac}(R)$ the fraction field, $L/K$ a finite galois extension, $R'$ the integral closure of $R$ in $L$. Then $R'$ is Dedekind again. Let $\mathfrak{p} \subset ...
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1answer
75 views

Why does $\mbox{Irr}(\alpha,K)$ have distinct roots in $N$?

My textbook assumes that $N\supset K$ is normal and finite $\alpha\in N$ is separable over $K$ $N\supset K(\alpha)$ is separable and deduces (in order to prove that $N\supset K$ is separable) that ...
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1answer
58 views

Degree of field extension question.

Last lecture it was written " $[\mathbb{Q}[X]:\mathbb{Q}]=\infty$. The elements $1,x,x^2,\ldots$ are linearly independent (but not a basis)." This confused me.. why isn't this a basis? Given any ...
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0answers
396 views

Bachelor Thesis - Galois Theory Research Topics?

I'm on the last semester of my bachelor's degree (undergrad degree) and I will be writing my thesis next semester. I have talked to a professor at my university and one of the topics he suggested was ...
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2answers
227 views

Applications of additive version of Hilbert's theorem 90

Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$, and $a$ is element of $L$ with trace 0, there exists an ...
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1answer
81 views

Integral Galois Extension (Serge Lang)

I have two questions about the proof of the following Proposition: Let $A$ be a ring, integrally closed in its quotient field $K$. Let $L$ be a finite Galois extension of $K$ with group $G$. Let $P$ ...
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2answers
57 views

My proof: $\alpha$ is NOT separable over $K\iff\mbox{Irr}(\alpha,K)'=0$

My proof goes as follows: Proof If $\alpha$ is a multiple root of $f=\newcommand{\Irr}{\mbox{Irr}(\alpha,K)}\Irr$ then $$ \begin{align} f&=(X-\alpha)^2g\\ f'&=2(X-\alpha)g+(X-\alpha)g' ...
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1answer
273 views

My proof: Frobenius Map generates $\mbox{Gal}(\mathbb F_{p^n}/\mathbb F_p)$

I would like to ask, whether anyone can confirm or correct the following version of the proof that the Frobenius map generates the Galois Group of a finite field. Proof First note that ...
4
votes
1answer
123 views

Is this proof that $\mathbb F_q^*$ is cyclic correct?

I would like to show that the multiplicative group of a finite field is cyclic. My goal is to use the best of my knowledge to render the argument as efficient yet simple and understandable as ...